Notions of (pointwise) tangential dimension are considered, for measures of R-N. Under regularity conditions (volume doubling), the upper resp. lower tangential dimension at a point x of a measure p can be defined as the supremum, resp. infimum, of local dimensions of the measures tangent to mu at x. Our main purpose is that of introducing a tool which is very sensitive to the "multifractal behaviour at a point" of a measure, namely which is able to detect the "oscillations" of the dimension at a given point, even when the local dimension exists, namely local upper and lower dimensions coincide. These definitions are tested on a class of fractals, which we call translation fractals, where they can be explicitly calculated for the canonical limit measure. In these cases the tangential dimensions of the limit measure coincide with the metric tangential dimensions of the fractal defined in [7], and they are constant, i.e. do not depend on the point. However, upper and lower dimensions may differ. Moreover, on these fractals, these quantities coincide with their noncommutative analogues, defined in previous papers [5, 6), in the framework of Alain Connes' noncommutative geometry.

Guido, D., Isola, T. (2006). Tangential dimensions II. Measures. HOUSTON JOURNAL OF MATHEMATICS, 32(2), 423-444.

Tangential dimensions II. Measures

GUIDO, DANIELE;ISOLA, TOMMASO
2006-01-01

Abstract

Notions of (pointwise) tangential dimension are considered, for measures of R-N. Under regularity conditions (volume doubling), the upper resp. lower tangential dimension at a point x of a measure p can be defined as the supremum, resp. infimum, of local dimensions of the measures tangent to mu at x. Our main purpose is that of introducing a tool which is very sensitive to the "multifractal behaviour at a point" of a measure, namely which is able to detect the "oscillations" of the dimension at a given point, even when the local dimension exists, namely local upper and lower dimensions coincide. These definitions are tested on a class of fractals, which we call translation fractals, where they can be explicitly calculated for the canonical limit measure. In these cases the tangential dimensions of the limit measure coincide with the metric tangential dimensions of the fractal defined in [7], and they are constant, i.e. do not depend on the point. However, upper and lower dimensions may differ. Moreover, on these fractals, these quantities coincide with their noncommutative analogues, defined in previous papers [5, 6), in the framework of Alain Connes' noncommutative geometry.
2006
Pubblicato
Rilevanza internazionale
Articolo
Sì, ma tipo non specificato
Settore MAT/05 - ANALISI MATEMATICA
English
Con Impact Factor ISI
tangent measure; translation fractals; METRIC-SPACES
http://www.math.uh.edu/~hjm/Vol32-2.html
Guido, D., Isola, T. (2006). Tangential dimensions II. Measures. HOUSTON JOURNAL OF MATHEMATICS, 32(2), 423-444.
Guido, D; Isola, T
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/31460
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